Magnetic resonance imaging relaxometry (MRIR) concerns the measurement of relaxation rates of spins that were excited by nuclear magnetic resonance (NMR). MRIR is based on the physical aspects of nuclei relaxing to the ground state after being excited by radio frequency (RF) energy associated with, for example, a spin inversion recovery sequence. To generate a “map” of relaxation rates (e.g., R2=1/T2) or of relaxation times (e.g., T1, T2), at least two magnetic resonance (MR) images are acquired. The sensitivity of the relaxometry technique depends on factors including, but not limited to, sequence, repetition time (TR), echo time (TE), the number of images acquired with different TE, the model adopted for fitting the acquired data, and so on. Conventional techniques have faced many challenges including accuracy and processing time.
MRIR is the process of recovering a map of relaxation parameters from the time sample of the spin signal for pixels in an MR image. Conventionally, MRIR has involved substantial computation that may have lead to unacceptably long parameter determination times. The long computation times have been associated with conventional mathematical approaches to solving the spin relaxometry inverse problem. In addition to taking a long time, the conventional approaches may have yielded inaccurate results.
Conventional mathematical techniques used in MRIR first considered that when an inversion recovery sequence is applied, the ideal value of a pixel's signal S at a time t is described by:S(t)=ρ(1−2e−xt)
where
ρ=spin density,
x=spin relaxation rate in sec−1, and
t=time in seconds.
However, since a pixel may represent several types of tissue, and since tissue types may have their own spin density (p) and their own spin relaxation rate (x), the ideal signal may better described by:
      S    ⁡          (      t      )        =            ∑      j        ⁢                  ρ        j            ⁡              (                  1          -                      2            ⁢                          ⅇ                              -                xjt                                                    )            
where j covers the different tissue types in the sample.
Even this representation of the S(t) equation may be unsatisfactory because this representation unnaturally assumes that relaxation rates for spins in tissue are unique and distinct with sharp demarcations. It is more likely that spins exhibiting a range of relaxation rates clustered around a central value(s) will be encountered. Therefore, spin density (p) may be more of a continuous function and less of a discrete function. When the continuous function approach is observed, then S(t) may be even more realistically described by:S(t)=∫ρ(x)(1−2e−xt)dx 
This representation would yield an idealized curve like curve 400 that is illustrated in FIG. 4. However, due to measurement noise and other factors, an actual input curve may be more like curve 500 that is illustrated in FIG. 5. The idealized curve 400 and the actual curve 500 illustrate that MRIR will involve processing the noisy MRI signal (e.g., curve 500) and deriving the spin spectrum that generated the signal. This is known as an inverse problem because it involves working backwards from an observed S(t) to determine the actual input ρ(x).
Attempts at solving the inverse problem for MRIR have been described as early as 1982. These early attempts report consuming up to 58 hours to solve the inverse problem for one 64×64 pixel image. While interesting from a theoretical mathematical perspective, 58 hours may not be a clinically relevant time frame. Thus, subsequent approaches attempted to speed up the calculations.
One subsequent approach involved applying parallel computing techniques to a Tikhonov regularization process for solving the inverse problem. Tikhonov regularization involves approximating a continuous integral as a discrete summation. Computations associated with the discrete summation may be parallelizable. Some subsequent attempts even progressed into constrained linear regularization. However, all of these subsequent approaches required some heuristic for selecting a trade off parameter for the regularization method. Ultimately, choosing a satisfactory trade-off parameter depended on having available the original spin spectrum for comparison. Unfortunately, in solving an MRIR inverse problem, the original spin spectrum is not available. Thus, computation times and accuracy continued to languish.
MRIR involves mapping the relaxation parameters for items (e.g., tissues) that have been excited using nuclear magnetic resonance (NMR). MRIR may produce relaxation parameter maps having pixel-wise parameter values for parameters including, but not limited to, T1 (spin-lattice relaxation), T2 (spin-spin relaxation), and spin density (M0) relaxation. Conventionally, these pixel-wise values may have been exploited in sub-optimal ways. One sub-optimal result was due to issues associated with performing conventional sliding window image reconstruction before performing relaxation parameter fitting. Another sub-optimal result was due to issues associated with performing conventional orthogonal matching pursuit (OMP) where dictionary entries were overly homogenous, particularly in the initial portion of a relaxation curve.
Recent advances in quantitative MRI data acquisition have facilitated, for under-sampled acquisitions, simultaneously determining multiple relaxation parameters but have yet to provide satisfactory treatment of the inverse problem. For example, Schmitt, et al., Inversion Recovery TrueFISP: Quantification of T1, T2, and Spin Density, Magn Reson Med 2004, 51:661-667, describe extracting multiple relaxation parameters from a signal time course sampled with a series of TrueFISP images after spin inversion. TrueFISP imaging refers to true fast imaging with steady state precession. TrueFISP is a coherent technique that uses a balanced gradient waveform. In TrueFISP, image contrast is determined by T2*/T1 properties mostly depending on repetition time (TR). As gradient hardware has improved, minimum TRs have been reduced. Additionally, as field shimming has improved, signal to noise ratio has improved making TrueFISP suitable for whole-body applications, for cardiac imaging, for brain tumor imaging, and for other applications. While this represents a significant advance in simultaneously acquiring pixel-wise values, the inverse problem remains an issue.
In another example, Twieg, Parsing local signal evolution directly from a single-shot MRI signal: a new approach for fMRI, Magn Reson Med 2003, November; 50(5):1043-52, describes a single-shot MRI method that performs single-shot parameter assessment by retrieval from signal encoding. Once again, data acquisition is improved, but the inverse problem remains. Doneva, et al., Compressed sensing reconstruction for magnetic resonance parameter mapping, Magn Reson Med 2010, Volume 64, Issue 4, pages 1114-1120 take one approach to the inverse problem. Doneva describes a dictionary based approach for parameter estimating. Doneva applies a learned dictionary to sparsify data and then uses a model based reconstruction for MR parameter mapping. Doneva identifies that “multiple relaxation components in a heterogeneous voxel can be assessed.” The success of this approach depends heavily on the learned dictionary. However, the Doneva library is limited to the idealized, single relaxation parameter curves because the preparation is specific and constrained by the fact that Doneva ultimately reconstructs an image from the acquired data. This constraint may yield a dictionary with overly homogenous relaxation curves against which input curves are to be fit or matched.
Yutzy S et al., Proc ISMRM 17 (2009), Pg. 2765 and Ehses P et al., Proc ISMRM 18 (2010) Pg. 2969, describe curve fitting based approaches for parameter estimating to solve the inverse problem. These approaches diverge from earlier approaches like those described by Carneiro, et al., MRI Relaxometry, Methods and Applications, Brazilian Journal of Physics, vol. 36, no. 1A, March 2006, which reported that “generally, T2 has been evaluated using single or bi-exponential fitting”, which lead to “the value of T2 obtained from an exponential fit [being] strongly influenced by the choice of the amplitude of the signal offset.” In some instances, both the single scale OMP and the bi-exponential fitting have yielded unsatisfactory results for the inverse problem.
Thus, even though more and more pixel-wise relaxation parameter data is becoming available, and even though that data is becoming available in ever shorter, more clinically relevant time frames, the data may still be under-utilized in conventional fitting approaches, even OMP approaches, that attempt to solve the inverse problem.
One conventional OMP technique may yield inaccurate results due to similarities between relaxation curves in the OMP dictionary. Recall that OMP involves matching or fitting acquired signal evolutions to dictionary relaxation curves to try to determine the best match or fit and thus to try to determine the relaxation parameter values. Conventionally, similarities between dictionary relaxation curves has made it difficult, if even possible at all, to identify a best match. In other words, when things in the dictionary are too similar, it is hard to use the dictionary to uniquely identify and disambiguate incoming signal evolutions because the incoming signal evolutions either match or do not match to the similar things in similar ways. This issue may be exacerbated when the incoming signal evolutions are associated with an under-sampled data space and have fewer points for comparison.
The short comings of conventional approaches have lead to a difficult decision concerning tradeoffs between high degrees of under-sampling and image quality. When items are relaxing very quickly, images may need to be highly under-sampled to catch fast relaxation components. However, the high under-sampling may lead to very noisy signal curves that are difficult to process and lead to lower quality images. However, when items are relaxing more slowly, high degrees of under-sampling may not be required and higher quality images may be acquired. Conventionally, imagers must balance the desire to catch fast relaxation components against the desire to have the highest quality images. With fewer images (e.g., higher under-sampling), the ability to track dynamic information is limited. Therefore, there is a tradeoff between image quality and the ability to track dynamic information. Using fewer projections allows making more images of lower quality but with higher ability to track dynamic information. Using more projections allows making fewer images of higher quality but with less ability to track dynamic information.